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Polynomial Methods: Recent Advancements in Combinatorics

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Author(s): Thomas Rexford Fleming
Series: master thesis at University of Memphis
Year: 2022

Language: English

Title
Abstract
Contents
1. Introduction and Preliminary Results
Introduction
Definitions, Notation, and Preliminary Results
2. Nullstellensatz
Hilbert’s Nullstellensatz
Alon’s Combinatorial Nullstellensatz
Ball and Serra’s Punctured Nullstellensatz
3. Simple Combinatorial Proofs
Sumsets
A Nice Result for Graphs
4. Zeilberger-Bressoud q-Dyson Theorem
The Dyson Conjecture and Lagrange Interpolation
The q-Dyson Theorem
5. Zero-Sum Theory
Definitions and Notation
Basic Proofs and Prerequisites
6. Davenport constant of finite abelian p-groups
Map Functors
Combinatorial Identities
The Davenport Constant
Remarks
7. The Chevalley-Warning Theorem and Reiher’s Proof of the Kemnitz Conjecture The Chevalley-Warning Theorem
The Chevalley-Warning Theorem
Reiher’s Proof of the Kemnitz Conjecture
8. Conclusion
References